# How do you write the partial fraction decomposition of the rational expression  x/((x+7)(x+8)(x+9))?

Aug 7, 2016

$\frac{x}{\left(x + 7\right) \left(x + 8\right) \left(x + 9\right)} = - \frac{7}{2 \left(x + 7\right)} + \frac{8}{x + 8} - \frac{9}{2 \left(x + 9\right)}$

#### Explanation:

$\frac{x}{\left(x + 7\right) \left(x + 8\right) \left(x + 9\right)} = \frac{A}{x + 7} + \frac{B}{x + 8} + \frac{C}{x + 9}$

Using Heaviside's cover-up method, we find:

$A = \frac{- 7}{\left(\left(- 7\right) + 8\right) \left(\left(- 7\right) + 9\right)} = \frac{- 7}{2} = - \frac{7}{2}$

$B = \frac{- 8}{\left(\left(- 8\right) + 7\right) \left(\left(- 8\right) + 9\right)} = \frac{- 8}{- 1} = 8$

$C = \frac{- 9}{\left(\left(- 9\right) + 7\right) \left(\left(- 9\right) + 8\right)} = \frac{- 9}{2} = - \frac{9}{2}$

So:

$\frac{x}{\left(x + 7\right) \left(x + 8\right) \left(x + 9\right)} = - \frac{7}{2 \left(x + 7\right)} + \frac{8}{x + 8} - \frac{9}{2 \left(x + 9\right)}$